Proof : Demonstration that a theorem is true. Axiom : A statement that is assumed to be true. Lemma : a less important theorem that is useful

Discrete Structures

Introduction to Proofs

Dr. Muhammad Humayoun

Assistant Professor

COMSATS Institute of Computer Science, Lahore.

Terminology

• Theorem: a statement that can be shown true. Sometimes called facts.

• Proof: Demonstration that a theorem is true.

• Axiom: A statement that is assumed to be true.

• Lemma: a less important theorem that is useful to prove a theorem.

• Corollary: a theorem that can be proven directly from a theorem that has been proved.

Stating Theorems

• Theorem. If 𝑥 > 𝑦, where x and y are positive real numbers, then 𝑥2 > 𝑦2.

• Theorem. For all positive real numbers x and

Trivial Proofs

• Consider an implication: 𝑝 → 𝑞

• If it can be shown that p is true, then the implication is always true

– By definition of an implication

Trivial Proof Example

Consider the statement:

• If you are in CSC102 then you are a student.

Vacuous proofs

• Consider an implication: 𝑝 → 𝑞

• If it can be shown that 𝑝 is false, then the implication is always true.

– By definition of an implication

Example

Consider the statement:

• Every student snooze during class when he is tired.

• Rephrased: If a student is tired then he snoozes during class

Methods of Proving Theorems

Methods of Proving Theorems

Direct Proofs

•

Consider an implication: 𝑝 → 𝑞

– If p is false, then the implication is always true

– Thus, show that if p is true, then q is true

Example Direct Proof

Theorem. Show that the square of an even number is an even number.

Example Direct Proof

Theorem. Show that the square of an even number is an even number.

For every number 𝑛, if 𝑛 is even, then 𝑛2 is even.

Example Direct Proof

Theorem. Show that the square of an even number is an even number.

For every number 𝑛, if 𝑛 is even, then 𝑛2 is even.

∀𝒏[even 𝒏 → even 𝒏𝟐 ] Proof.

Usual convention: Universal instantiation is not explicitly used and it is directly showed that

Example Direct Proof

Theorem. Show that the square of an even number is an even number.

For every number 𝑛, if 𝑛 is even, then 𝑛2 is even.

∀𝒏[even 𝒏 → even 𝒏𝟐 ]

Proof.

Usual convention: Universal instantiation is not explicitly used and it is directly showed that even(𝑛) implies even(𝑛2).

Assume 𝑛 is even.

Thus, 𝑛 = 2𝑘, for some 𝑘 (definition of even numbers).

Example Direct Proof

Theorem. Show that the square of an even number is an even number.

For every number 𝑛, if 𝑛 is even, then 𝑛2 is even.

∀𝒏[even 𝒏 → even 𝒏𝟐 ]

Proof.

Usual convention: Universal instantiation is not explicitly used and it is directly showed that even(𝑛) implies even(𝑛2).

Assume 𝑛 is even.

Proof by Contraposition

(or Indirect proofs)

• 𝑝 → 𝑞 ≡ ¬𝑞 → ¬𝑝

– If the antecedent (¬𝑞) is false, then the contrapositive is always true

– Thus, show that if ¬𝑞 is true, then ¬𝑝 is true

Indirect proof example

Theorem. If 𝑛2 is an odd integer then 𝑛 is an odd integer

Proof (by contrapositive).

We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold.

Contrapositive: If 𝑛 is an even integer, then 𝑛2 is an even integer.

Indirect proof example

Theorem. If 𝑛2 is an odd integer then 𝑛 is an odd integer

Proof (by contrapositive).

We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold.

Contrapositive: If 𝑛 is an even integer, then 𝑛2 is an even integer.

Indirect proof example

Theorem. If 𝑛2 is an odd integer then 𝑛 is an odd integer

Proof (by contrapositive).

We show that the contrapositive of the theorem statement holds, therefore the theorem statement hold.

Contrapositive: If 𝑛 is an even integer, then 𝑛2 is an even integer.

Assume that 𝑛 is even. Then by the definition of even numbers: 𝑛 = 2𝑘 for some integer 𝑘. Thus

Selecting a Proof method

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then

𝑛 is even.

Proof. (Via direct proof). Assume that 𝑛3 + 5 is odd.

Then by the definition of odd numbers 𝑛3 + 5 =

2𝑘 + 1 for some integer k. Thus

𝑛3 = 2𝑘 − 4

Selecting a Proof method

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then

𝑛 is even.

Proof. (Via direct proof). Assume that 𝑛3 + 5 is odd.

Then by the definition of odd numbers 𝑛3 + 5 =

2𝑘 + 1 for some integer k. Thus

𝑛3 = 2𝑘 − 4

Selecting a Proof method

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Indirect proof.

Contrapositive: If 𝑛 is odd, then 𝑛3 + 5 is even. Assume 𝑛 is odd, and show that 𝑛3 + 5 is even.

By the definition of odd numbers 𝑛 = 2𝑘 + 1 for some integer 𝑘.

𝑛3 + 5 = 2𝑘 + 1 3 + 5 = 8𝑘3 + 12𝑘2 + 6𝑘 + 6 = 2(4𝑘3 + 6𝑘2 + 3𝑘 + 3)

Selecting a Proof method

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Indirect proof.

Contrapositive: If 𝑛 is odd, then 𝑛3 + 5 is even.

Assume 𝑛 is odd, and show that 𝑛3 + 5 is even.

By the definition of odd numbers 𝑛 = 2𝑘 + 1 for some integer 𝑘.

𝑛3 + 5 = 2𝑘 + 1 3 + 5 = 8𝑘3 + 12𝑘2 + 6𝑘 + 6 = 2(4𝑘3 + 6𝑘2 + 3𝑘 + 3)

Selecting a Proof method

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Indirect proof.

Contrapositive: If 𝑛 is odd, then 𝑛3 + 5 is even. Assume 𝑛 is odd, and show that 𝑛3 + 5 is even.

By the definition of odd numbers 𝑛 = 2𝑘 + 1 for some integer 𝑘.

𝑛3 + 5 = 2𝑘 + 1 3 + 5 = 8𝑘3 + 12𝑘2 + 6𝑘 + 6 = 2(4𝑘3 + 6𝑘2 + 3𝑘 + 3)

Selecting a Proof method

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Indirect proof.

Contrapositive: If 𝑛 is odd, then 𝑛3 + 5 is even. Assume 𝑛 is odd, and show that 𝑛3 + 5 is even.

By the definition of odd numbers 𝑛 = 2𝑘 + 1 for some integer 𝑘.

𝑛3 + 5 = 2𝑘 + 1 3 + 5 = 8𝑘3 + 12𝑘2 + 6𝑘 + 6 = 2(4𝑘3 + 6𝑘2 + 3𝑘 + 3)

Proof by contradiction

•

Given a statement of the form

𝑝 → 𝑞

•

Assume

p is true

and

q is false

– Assume p and assume ¬𝑞

•

Then prove that

¬𝑞

cannot occur

Proof by contradiction example

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Rephrased: If 𝑛3 + 5 is odd, then 𝑛 is even

Proof. Assume p is true and q is false (Assume 𝑝. Assume ¬𝑞)

Proof by contradiction example

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Rephrased: If 𝑛3 + 5 is odd, then 𝑛 is even

Proof. Assume p is true and q is false (Assume 𝑝. Assume ¬𝑞)

Assume that 𝑛3 + 5 is odd, and 𝑛 is odd.

Proof by contradiction example

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even. Rephrased: If 𝑛3 + 5 is odd, then 𝑛 is even

Proof. Assume p is true and q is false (Assume 𝑝. Assume ¬𝑞)

Assume that 𝑛3 + 5 is odd, and 𝑛 is odd.

By the definition of odd numbers 𝑛 = 2𝑘 + 1 for some integer k.

Proof by contradiction example

Theorem. If 𝑛 is an integer and 𝑛3 + 5 is odd, then 𝑛 is even.

Rephrased: If 𝑛3 + 5 is odd, then 𝑛 is even

Proof. Assume p is true and q is false (Assume 𝑝. Assume ¬𝑞)

Assume that 𝑛3 + 5 is odd, and 𝑛 is odd.

By the definition of odd numbers 𝑛 = 2𝑘 + 1 for some integer

k.

𝑛3 + 5 = 2𝑘 + 1 3 + 5 = 8𝑘3 + 12𝑘2 + 6𝑘 + 6 = 2(4𝑘3 + 6𝑘2 + 3𝑘 + 3)